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An equivalence between high dimensional Bayes optimal inference and M-estimation

Due to the computational difficulty of performing MMSE (minimum mean squared error) inference, maximum a posteriori (MAP) is often used as a surrogate. However, the accuracy of MAP is suboptimal for high dimensional inference, where the number of model parameters is of the same order as the number of samples. In this work we demonstrate how MMSE performance is asymptotically achievable via optimization with an appropriately selected convex penalty and regularization function which are a smoothed version of the widely applied MAP algorithm. Our findings provide a new derivation and interpretation for recent optimal M-estimators discovered by El Karoui, et. We demonstrate the performance of these optimal M-estimators with numerical simulations.

Adversarial Risk via Optimal Transport and Optimal Couplings

The accuracy of modern machine learning algorithms deteriorates severely on adversarially manipulated test data. Optimal adversarial risk quantifies the best error rate of any classifier in the presence of adversaries, and optimal adversarial classifiers are sought that minimize adversarial risk. In this paper, we investigate the optimal adversarial risk and optimal adversarial classifiers from an optimal transport perspective. We present a new and simple approach to show that the optimal adversarial risk for binary classification with $0-1$ loss function is completely characterized by an optimal transport cost between the probability distributions of the two classes, for a suitably defined cost function. We propose a novel coupling strategy that achieves the optimal transport cost for several univariate distributions like Gaussian, uniform and triangular. Using the optimal couplings, we obtain the optimal adversarial classifiers in these settings and show how they differ from optimal classifiers in the absence of adversaries. Based on our analysis, we evaluate algorithm-independent fundamental limits on adversarial risk for CIFAR-10, MNIST, Fashion-MNIST and SVHN datasets, and Gaussian mixtures based on them. In addition to the $0-1$ loss, we also derive bounds on the deviation of optimal risk and optimal classifier in the presence of adversaries for continuous loss functions, that are based on the convexity and smoothness of the loss functions.

Fast and Memory Optimal Low-Rank Matrix Approximation

In this paper, we revisit the problem of constructing a near-optimal rank $k$ approximation of a matrix $M\in [0,1] {m\times n}$ under the streaming data model where the columns of $M$ are revealed sequentially. We present SLA (Streaming Low-rank Approximation), an algorithm that is asymptotically accurate, when $k s_{k 1} (M) o(\sqrt{mn})$ where $s_{k 1}(M)$ is the $(k 1)$-th largest singular value of $M$. This means that its average mean-square error converges to 0 as $m$ and $n$ grow large (i.e., $\ \hat{M} {(k)}-M {(k)} \ _F 2 o(mn)$ with high probability, where $\hat{M} {(k)}$ and $M {(k)}$ denote the output of SLA and the optimal rank $k$ approximation of $M$, respectively). Our algorithm makes one pass on the data if the columns of $M$ are revealed in a random order, and two passes if the columns of $M$ arrive in an arbitrary order. To reduce its memory footprint and complexity, SLA uses random sparsification, and samples each entry of $M$ with a small probability $\delta$.

k*-Nearest Neighbors: From Global to Local

The weighted k-nearest neighbors algorithm is one of the most fundamental non-parametric methods in pattern recognition and machine learning. The question of setting the optimal number of neighbors as well as the optimal weights has received much attention throughout the years, nevertheless this problem seems to have remained unsettled. In this paper we offer a simple approach to locally weighted regression/classification, where we make the bias-variance tradeoff explicit. Our formulation enables us to phrase a notion of optimal weights, and to efficiently find these weights as well as the optimal number of neighbors efficiently and adaptively, for each data point whose value we wish to estimate. The applicability of our approach is demonstrated on several datasets, showing superior performance over standard locally weighted methods.

Wasserstein Variational Gradient Descent: From Semi-Discrete Optimal Transport to Ensemble Variational Inference

Particle-based variational inference offers a flexible way of approximating complex posterior distributions with a set of particles. In this paper we introduce a new particle-based variational inference method based on the theory of semi-discrete optimal transport. Instead of minimizing the KL divergence between the posterior and the variational approximation, we minimize a semi-discrete optimal transport divergence. The solution of the resulting optimal transport problem provides both a particle approximation and a set of optimal transportation densities that map each particle to a segment of the posterior distribution. We approximate these transportation densities by minimizing the KL divergence between a truncated distribution and the optimal transport solution. The resulting algorithm can be interpreted as a form of ensemble variational inference where each particle is associated with a local variational approximation.